Isoperimetric Inequalities

Brunn-Minkowski理论，通常被称为混合体理论，是凸几何分析的核心。多年来，数学、科学和工程的其他领域已经发现了许多应用--或与之联系--，包括偏微分方程组、代数几何、统计学、数论、明可夫斯基和芬斯勒几何、体视学、信息论。这个提议的目的是发展经典的Brunn-Minkowski理论的扩展和对偶。一个特别的焦点是仿射等周不等式的发展。仿射不等式在偏微分方程组、Banach空间几何、几何层析成像甚至机器人视觉中都有大量的应用。另一个特别的焦点将是仿射等周不等式的解析对应的发展。PI开发的新方法将被用来试图在一些长期猜测的仿射等周不等式上取得进展。该提案的另一个主要焦点涉及规定的曲率问题，称为LP Minkowski问题，它自然地出现在扩展的Brunn-Minkowski理论中。PI的工作表明，信息论和凸几何分析之间存在着有趣的潜在联系。经典的Brunn-Minkowski理论提供了解决各种基本逆问题所需的强大工具，其中唯一可用的“数据”涉及凸体在低维子空间(如直线和平面)上的投影信息。然而，经典的Brunn-Minkowski理论的工具在处理这些问题的对偶方面几乎没有价值，其中“对子空间的投影”被“有子空间的交集”所取代。私人投资机构正在继续努力发展布鲁恩-明科夫斯基对偶理论。这种对偶理论已经解决了一个长期存在的问题，即已知信息涉及未知物体与平面的交点。这种“逆问题”不仅是数学的基础，也是科学和工程的基础。

The Brunn-Minkowski theory, often called the theory of mixed volumes, is the very core of convex geometric analysis. Over the years numerous applications to---or connections with---other fields of mathematics, science, and engineering have been discovered, including partial differential equations, algebraic geometry, statistics, number theory, Minkowski and Finsler geometry, stereology, information theory. The goal of this proposal is the development of extensions and duals of the classical Brunn-Minkowski theory. A special focus is the development of affine isoperimetric inequalities. There have been numerous applications of affine inequalities in PDEs, Banach space geometry, geometric tomography, and even robot vision. Another special focus will be the development of the analytic counterparts of affine isoperimetric inequalities. New methods developed by the PIs will be used to attempt to make progress on a number of long-conjectured affine isoperimetric inequalities. Another main focus of the proposal concerns a prescribed curvature problem, known as the Lp Minkowski problem, that arises naturally within the extended Brunn-Minkowski theory. Work of the PIs indicates that there are interesting potential connections between information theory and convex geometric analysis. These connections will be exploited with the aim of establishing new geometric results inspired by their information theory counterparts.The classical Brunn-Minkowski theory has provided powerful tools needed to solve a variety of basic inverse problems where the only "data" available involves information regarding the projections of convex bodies onto lower-dimensional subspaces (such as lines and planes). However, the tools of the classical Brunn-Minkowski theory have been of little value in dealing with the dual of these questions, where "projections onto subspaces" are replaced by "intersections with subspaces". The PIs are continuing efforts to develop a dual Brunn-Minkowski theory. This dual theory has already led to the solution of a longstanding problem where the known information involved the intersections of the unknown bodies with planes. Such "inverse problems" are basic in not only mathematics but also science and engineering.